On the trivariate joint distribution of Brownian motion and its maximum and minimum

The trivariate joint probability density function of Brownian motion and its maximum and minimum can be expressed as an infinite series of normal probability density functions. In this letter, we show that the infinite series converges uniformly, and satisfies the Fokker–Planck equation. Also, we express it as a product form using Jacobi’s triple product identity, and present some error bounds of a finite series approximation of the infinite series.